Sunday, August 28

A factorization balancing act

A couple of weeks ago, Claudio Meller presented 26487 and 65821 as examples of the property of having one each of the base-ten digits when combined with the digits of their respective factorizations. Surprisingly, he missed two:

I wondered how this might be turned into a sequence. Base-ten k-balanced factorization integers: The combined digits of an integer and its factorization primes and exponents contain exactly k copies of each of the ten digits. So,

45849660 = 2^2 * 3 * 5 * 19 * 37 * 1087

84568740 = 2^2 * 3 * 5 * 67 * 109 * 193

104086845 = 3^2 * 5 * 19 * 23 * 67 * 79

106978404 = 2^2 * 3 * 13 * 685759

and so on. For any given k, k-balanced integers are necessarily finite. For k=2, there are a little over 13000. Is the largest of these greater than the smallest 3-balanced integer?